We have seen that \(\widetilde{\mathcal {R}}_{\mathrm {c},\kappa }\)  is a factor. What remains is to show that \(\widetilde{\mathcal {R}}_{\mathrm {c},\kappa }\)  contains
a minimal projection. As in Lemma REF  we take implementing unitary
\(u(\kappa ) \in \mathcal {R}_{\mathrm {c}}\)  for \({\hbox{\rm Ad\,}}V_{\mathrm {c}}(\kappa )\) , this time indicating the dependence on \(\kappa \) .
By a classical result by Bargmann, we may assume that \(u(\kappa )\)  is a one-parameter
group of unitaries [1]}, where our group is \(S^1\)  hence
there is no nontrivial antisymmetric bilinear form on its Lie algebra \({\mathbb {R}}\) .
